Grimoire for the Apprentice Wizard

274 Grimoire for the Apprentice Wizard

The Golden Mean

One of the most important numerical relationships

noted by the ancient Greek geometricians was what

they called The Golden Mean. This refers to a ratio of

1.618034... rounded to 1.62—a number called Phi

(Φ). The Golden Mean is related to the Fibonacci se-

quence. For an explanation, take the ratios of the fol-

lowing successive numbers from the Fibonacci series:

1/1 = 1 13/8 = 1.625

2/1 = 2 21/13 = 1.615

3/2 = 1.5 34/21 = 1.619

5/3 = 1.66 55/34 = 1.617

8/5 = 1.60 89/55 = 1.618

As the Fibonacci numbers get larger, their ratio ap-

proaches the 1.62 ratio of the Golden Mean. This is

not surprising if we look at the Golden Rectangle, in

which the side of every square is equal to the sum of

the sides of the next two squares. This is same con-

cept that determines the Fibonacci sequence.

The Golden Mean (also called Golden Ratio, Phi

Ratio, Sacred Cut, and Divine Proportion) is a fun-

damental measure that seems to crop up almost every-

where, including crops. It also governs all body pro-

portions, such the respective lengths of your fingers,

upper leg to lower, etc. (The actual ratio is about

1.618033988749894848204586834365638117720309180....)

The Golden Ratio is the unique proportion such that

the ratio of the whole to the larger portion is the same

as the ratio of the larger portion to the smaller. As

such, it symbolically links each new generation to its

ancestors, preserving the continuity of relationship as

the means for retracing its lineage.

As scholars and artists of eras gone by discov-

ered, the intentional use of these natural proportions in

art of various forms expands our sense of beauty, bal-

ance, and harmony to optimal effect. The most famous

building of Classical Greece, and one of the Seven

Wonders of the ancient world, was the Parthenon in the

Acropolis at Athens. Its proportions are all based on

the Golden Rectangle. Other buildings in ancient

Greece contain a similar height to length ratio.

The Golden Spiral

We can make another picture showing the Fi-

bonacci Sequence if we start with two small squares

of size 1 next to each other. Above of both of these

draw a square of size 2 (=1+1).

We can now draw a new square—touching both

a unit square and the latest square of side 2—so hav-

ing sides 3 units long; and then another touching both

the 2-square and the 3-square (which has sides of 5

units). We can continue adding squares around the pic-

ture, each new square having a side which is as long as

the sum of the last two squares’ sides. This set of rect-

angles whose sides are two successive Fibonacci num-

bers in length, and which are composed of squares

with sides which are Fibonacci numbers, are called Fi-

bonacci Rectangles. However many you make this

way, they always add up to a Golden Rectangle.

Now we can draw a spiral by putting together

quarter circles, one in each new square. This is the

Fibonacci Spiral. A similar curve to this occurs in na-

ture as the shape of a snail shell or some seashells.

Whereas the Fibonacci Rectangles spiral increases in

size by a factor of Phi (1.62) in a quarter of a turn (i.e.,

a point a further quarter of a turn round the curve is

1.62 times as far from the centre, and this applies to all

points on the curve), the spiral curve of a chambered

nautilus shell takes a whole turn before points move a

factor of 1.62 from the centre.

We can also see similar spirals from the atomic

Golden

Rectangle

The Parthenon

ΦΦΦΦΦ

Fibonacci

Rectangles

with

Golden Spiral

Chambered

Nautilus

2

3

(^85)
13 21

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